5 11 13 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 11   c = 13

Area: T = 26.89221456935
Perimeter: p = 29
Semiperimeter: s = 14.5

Angle ∠ A = α = 22.09331769923° = 22°5'35″ = 0.38655986807 rad
Angle ∠ B = β = 55.83877404834° = 55°50'16″ = 0.97545524183 rad
Angle ∠ C = γ = 102.0699082524° = 102°4'9″ = 1.78114415545 rad

Height: ha = 10.75768582774
Height: hb = 4.88994810352
Height: hc = 4.13772531836

Median: ma = 11.77992189894
Median: mb = 8.17700673192
Median: mc = 5.54552682532

Inradius: r = 1.85546307375
Circumradius: R = 6.6476922192

Vertex coordinates: A[13; 0] B[0; 0] C[2.80876923077; 4.13772531836]
Centroid: CG[5.26992307692; 1.37990843945]
Coordinates of the circumscribed circle: U[6.5; -1.39898110038]
Coordinates of the inscribed circle: I[3.5; 1.85546307375]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.9076823008° = 157°54'25″ = 0.38655986807 rad
∠ B' = β' = 124.1622259517° = 124°9'44″ = 0.97545524183 rad
∠ C' = γ' = 77.93109174756° = 77°55'51″ = 1.78114415545 rad

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How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 5 ; ; b = 11 ; ; c = 13 ; ;

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 5+11+13 = 29 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29 }{ 2 } = 14.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.5 * (14.5-5)(14.5-11)(14.5-13) } ; ; T = sqrt{ 723.19 } = 26.89 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 26.89 }{ 5 } = 10.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 26.89 }{ 11 } = 4.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 26.89 }{ 13 } = 4.14 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 5**2-11**2-13**2 }{ 2 * 11 * 13 } ) = 22° 5'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11**2-5**2-13**2 }{ 2 * 5 * 13 } ) = 55° 50'16" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-5**2-11**2 }{ 2 * 11 * 5 } ) = 102° 4'9" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 26.89 }{ 14.5 } = 1.85 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 22° 5'35" } = 6.65 ; ;




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